Some questions on blowups, strict transforms and its deformation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:15:52Z http://mathoverflow.net/feeds/question/67277 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67277/some-questions-on-blowups-strict-transforms-and-its-deformation Some questions on blowups, strict transforms and its deformation Dmitry Kerner 2011-06-08T16:09:41Z 2011-06-08T16:09:41Z <p>Let $Z\subseteq Sing(X)\subset M$, where all objects are complex reduced and projective, moreover $Z$ and $M$ are smooth. Consider the strict transform <code>$\tilde{X}\subset Bl_Z M$</code>.</p> <ol> <li><p>If all the singularities of $X$ are of hypersurface type then so are all the singularities of $\tilde{X}$. I guess this is true even if the (smooth) center of blowup lies in $X$ but is not contained in $Sing(X)$. Is the same true for locally complete intersections (at least for the case $Z\subseteq Sing(X)$)? Is there some even bigger class of singularities that is preserved under blowups with smooth centers? (something more restrictive than Cohen-Macaulay) Related questions have been asked here (<a href="http://mathoverflow.net/questions/43336/blowups-of-cohen-macaulay-varieties" rel="nofollow">1</a>, <a href="http://mathoverflow.net/questions/62396/when-is-a-blow-up-cohen-macaulay" rel="nofollow">2</a>) but I can't find an answer there :(</p></li> <li><p>Let <code>$(X,0)=\{f=0\}\subset(\Bbb C^N,0)$</code> be an isolated hypersurface singularity of multiplicity $p$. The strict transform $\tilde{X}\subset Bl_0(\Bbb C^N,0)$ admits a very specific smoothing that comes from downstairs: <code>$(X_{\epsilon},0)=\{f+\epsilon (x^p_1+..+x^p_n)=0\}$</code>. I.e. the (flat) equi-multiple family <code>$(X_{\epsilon},0)$</code> singularities that possesses simultaneous strict transform. "Some sort of smoothing with descent". I guess the same is true for locally complete intersections, right? Is it true in some more general situation?</p></li> <li><p>Suppose $X\subset M$ is a hypersurface, the zero locus of a section $s\in\Gamma(\mathcal{L}_M)$. Let $p=mult_Z X$ be the generic multiplicity. Then the strict transform $\tilde{X}\subset Bl_ZM$ is the zero locus of a section of the bundle <code>$\pi^*\mathcal{L}_M(-pE)$</code>. One can write a similar formula for <code>$X\subset M$</code> a complete intersection. What about a more general case?</p></li> </ol>